# Equivalence relation

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Let $S$ be a set. A binary relation $\sim$ on $S$ is said to be an equivalence relation if $\sim$ satisfies the following three properties:

1. For every element $x \in S$, $x \sim x$. (Reflexive property)

2. If $x, y \in S$ such that $x \sim y$, then we also have $y \sim x$. (Symmetric property)

3. If $x, y, z \in S$ such that $x \sim y$ and $y \sim z$, then we also have $x \sim z$. (Transitive property)

Some common examples of equivalence relations:

• The relation $=$ (equality), on the set of real numbers.
• The relation $\cong$ (congruence), on the set of geometric figures in the plane.
• The relation $\sim$ (similarity), on the set of geometric figures in the plane.
• For a given positive integer $n$, the relation $\equiv \pmod n$, on the set of integers. (Congruence mod n)